Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions

We consider the deformation of the Poincaré group in 2+1 dimensions into the quantum double of the Lorentz group and construct Lorentz-covariant momentum-space formulations of the irreducible representations describing massive particles with spin 0, $\frac12$ and 1 in the deformed theory. We discuss ways of obtaining non-commutative versions of relativistic wave equations like the Klein–Gordon, Dirac and Proca equations in 2+1 dimensions by applying a suitably defined Fourier transform, and point out the relation between non-commutative Dirac equations and the exponentiated Dirac operator considered by Atiyah and Moore.